Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Show that if $A$ and $B$ are two subsets of $\mathbb{R}^n$ and $\gamma \in \mathbb{Z}^n$ we have $$E_{A \cup B} + E_{A \cap B} = E_A + E_B \quad \text{and} \quad E_{\gamma + A} = x^\gamma E_A.$$

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$, and we are given $f \in \mathcal{C}^1(\mathbb{R}^d)$. Let $x_*$ be a minimizer of $f$ on $C$. Suppose in this question that $\|x_*\| < 1$. Show that $\nabla f(x_*) = 0$.

12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.

Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.

1	2	3	4	5	6	7	8
9	10	11	12	13	14	15	16
17	18	19	20	21	22	23	24
25	26	27	28	29	30	31	32
33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48
49	50	51	52	53	54	55	56
57	58	59	60	61	62	63	64

Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.

1	2	3	4	5	6	7	8
9	10	11	12	13	14	15	16
17	18	19	20	21	22	23	24
25	26	27	28	29	30	31	32
33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48
49	50	51	52	53	54	55	56
57	58	59	60	61	62	63	64

Let $k , n$ and $r$ be positive integers.
(a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying
$$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$
(b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.

Let $ABCD$ be a quadrilateral with all internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta _ { 1 } , \Delta _ { 2 } , \Delta _ { 3 }$ and $\Delta _ { 4 }$ denote the areas of the shaded triangles shown. Prove that
$$\Delta _ { 1 } - \Delta _ { 2 } + \Delta _ { 3 } - \Delta _ { 4 } = 0 .$$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function which is differentiable at 0. Define another function $g : \mathbb { R } \rightarrow \mathbb { R }$ as follows:
$$g ( x ) = \begin{cases} f ( x ) \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{cases}$$
Suppose that $g$ is also differentiable at 0. Prove that
$$g ^ { \prime } ( 0 ) = f ^ { \prime } ( 0 ) = f ( 0 ) = g ( 0 ) = 0$$

Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.

Let $a , b , c$ be nonzero real numbers such that $a + b + c \neq 0$. Assume that $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = \frac { 1 } { a + b + c }$$ Show that for any odd integer $k$, $$\frac { 1 } { a ^ { k } } + \frac { 1 } { b ^ { k } } + \frac { 1 } { c ^ { k } } = \frac { 1 } { a ^ { k } + b ^ { k } + c ^ { k } }$$

Let $ABC$ be a right triangle with right angle at $A$. Let $O$ be the center of the square $BCDE$ constructed on the hypotenuse, on the opposite side from vertex $A$.
Prove that $O$ is equidistant from the lines $AB$ and $AC$.

6. In a Cartesian coordinate system $O x y$, consider the equilateral hyperbola with equation $x y = k$, with $k$ a non-zero real parameter. Let $t$ be the tangent line to the hyperbola at a point $P$ of it. Let $A$ and $B$ be the points where $t$ intersects the axes of the reference frame. Prove that the triangles $A P O$ and $B P O$ are equivalent and that their area does not depend on the choice of $P$.

1. Given a triangle $A B C$, let $M$ be the midpoint of side $B C$ and let $B ^ { \prime }$ and $C ^ { \prime }$ be two points, respectively, on side $A B$ and on side $A C$, such that $A B ^ { \prime } = \frac { 1 } { 3 } A B$ and $A C ^ { \prime } = \frac { 1 } { 3 } A C$. Prove that, if the segments $M B ^ { \prime }$ and $M C ^ { \prime }$ are congruent to each other, then so are the sides $A B$ and $A C$.

If In is the area of $n$ sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that $$I _ { n } = \frac { O _ { n } } { 2 } \left( 1 + \sqrt { 1 - \left( \frac { 2 l _ { n } } { n } \right) ^ { 2 } } \right)$$

The negation of $\sim s \vee (\sim r \wedge s)$ is equivalent to:
(1) $s \wedge \sim r$
(2) $s \wedge (r \wedge \sim s)$
(3) $s \vee (r \vee \sim s)$
(4) $s \wedge r$

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to:
(1) $p \wedge q$
(2) $p \vee q$
(3) $p \vee \sim q$
(4) $\sim p \wedge q$

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is:
(1) If the area of a square increases four times, then its side is not doubled.
(2) If the area of a square does not increase four times, then its side is not doubled.
(3) If the area of a square does not increase four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.

The Boolean expression $(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$ is equivalent to: (1) $\sim p \wedge q$ (2) $p \wedge q$ (3) $p \vee q$ (4) $p \vee \sim q$

The contrapositive of the following statement, ``If the side of a square doubles, then its area increases four times'', is: (1) If the area of a square increases four times, then its side is not doubled. (2) If the area of a square does not increase four times, then its side is not doubled. (3) If the area of a square does not increase four times, then its side is doubled. (4) If the side of a square is not doubled, then its area does not increase four times.

For two statements $p$ and $q$, the logical statement $(p \rightarrow q) \wedge (q \rightarrow \sim p)$ is equivalent to
(1) $p$
(2) $q$
(3) $\sim p$
(4) $\sim q$

Let $A , B , C$ and $D$ be four non-empty sets. The contrapositive statement of "If $A \subseteq B$ and $B \subseteq D$, then $A \subseteq C$" is
(1) If $A \nsubseteq C$, then $A \subseteq B$ and $B \subseteq D$
(2) If $A \subseteq C$, then $B \subset A$ and $D \subset B$
(3) If $A \nsubseteq C$, then $A \nsubseteq B$ and $B \subseteq D$
(4) If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$

Negation of the statement: $\sqrt { 5 }$ is an integer or 5 is irrational is:
(1) $\sqrt { 5 }$ is not an integer 5 is not irrational
(2) $\sqrt { 5 }$ is not an integer and 5 is not irrational
(3) $\sqrt { 5 }$ is irrational or 5 is an integer
(4) $\sqrt { 5 }$ is an integer and 5 irrational

The contrapositive of the statement ``If I reach the station in time, then I will catch the train'' is
(1) If I do not reach the station in time, then I will catch the train.
(2) If I do not reach the station in time, then I will not catch the train.
(3) If I will catch the train, then I reach the station in time.
(4) If I will not catch the train, then I do not reach the station in time.

The proposition $p \rightarrow \sim ( p \wedge \sim q )$ is equivalent to:
(1) $q$
(2) $( \sim p ) \vee q$
(3) $( \sim p ) \wedge q$
(4) $( \sim p ) \vee ( \sim q )$

Contrapositive of the statement: 'If a function $f$ is differentiable at $a$, then it is also continuous at $a$', is
(1) If a function $f$ is continuous at $a$, then it is not differentiable at $a$.
(2) If a function $f$ is not continuous at $a$, then it is not differentiable at $a$.
(3) If a function $f$ is not continuous at $a$, then it is differentiable at $a$.
(4) If a function $f$ is continuous at $a$, then it is differentiable at $a$.